Average Error: 29.8 → 0.0
Time: 15.2s
Precision: 64
Internal Precision: 1344
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006528557608830371 \lor \neg \left(x \le 0.00677807468883077\right):\\ \;\;\;\;\frac{\log \left(e^{(\left(\frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1}\right) + -1)_*}\right)}{1 + \sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \frac{2}{e^{-2 \cdot x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.006528557608830371 or 0.00677807468883077 < x

    1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]
    4. Using strategy rm

    5. Applied add-log-exp0.0

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - \color{blue}{\log \left(e^{1 \cdot 1}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    6. Applied add-log-exp0.0

      \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}}\right)} - \log \left(e^{1 \cdot 1}\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    7. Applied diff-log0.0

      \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}}}{e^{1 \cdot 1}}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

    8. Simplified0.0

      \[\leadsto \frac{\log \color{blue}{\left(e^{(\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}}\right) + -1)_*}\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
    9. Using strategy rm

    10. Applied add-cbrt-cube0.0

      \[\leadsto \frac{\log \left(e^{(\left(\frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}}\right) + -1)_*}\right)}{\color{blue}{\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}}} + 1}\]

    if -0.006528557608830371 < x < 0.00677807468883077

    1. Initial program 59.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006528557608830371 \lor \neg \left(x \le 0.00677807468883077\right):\\ \;\;\;\;\frac{\log \left(e^{(\left(\frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \left(\frac{2}{e^{-2 \cdot x} + 1}\right) + -1)_*}\right)}{1 + \sqrt[3]{\left(\frac{2}{e^{-2 \cdot x} + 1} \cdot \frac{2}{e^{-2 \cdot x} + 1}\right) \cdot \frac{2}{e^{-2 \cdot x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} \cdot \frac{2}{15} + x\right) - {x}^{3} \cdot \frac{1}{3}\\ \end{array}\]

Runtime

Time bar (total: 15.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.80.00.029.8100%
herbie shell --seed 2018295 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))